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Properties

Label 2-9408-1.1-c1-0-139
Degree 22
Conductor 94089408
Sign −1-1
Analytic cond. 75.123275.1232
Root an. cond. 8.667368.66736
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 2·11-s + 13-s + 2·17-s − 5·19-s − 6·23-s − 5·25-s + 27-s + 8·29-s − 3·31-s − 2·33-s + 9·37-s + 39-s − 2·41-s + 43-s + 8·47-s + 2·51-s − 6·53-s − 5·57-s − 6·59-s − 2·61-s − 5·67-s − 6·69-s − 4·71-s + 11·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.277·13-s + 0.485·17-s − 1.14·19-s − 1.25·23-s − 25-s + 0.192·27-s + 1.48·29-s − 0.538·31-s − 0.348·33-s + 1.47·37-s + 0.160·39-s − 0.312·41-s + 0.152·43-s + 1.16·47-s + 0.280·51-s − 0.824·53-s − 0.662·57-s − 0.781·59-s − 0.256·61-s − 0.610·67-s − 0.722·69-s − 0.474·71-s + 1.28·73-s − 0.577·75-s + ⋯

Functional equation

Λ(s)=(9408s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9408s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 94089408    =    26⋅3⋅722^{6} \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 75.123275.1232
Root analytic conductor: 8.667368.66736
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9408, ( :1/2), −1)(2,\ 9408,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1−T 1 - T
7 1 1
good5 1+pT2 1 + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−9T+pT2 1 - 9 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+5T+pT2 1 + 5 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1+18T+pT2 1 + 18 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.60402456637185343398224581455, −6.62447757568286724337784834863, −6.06709869671519499992746204907, −5.34443408102195224933515059274, −4.33471889531552356021563552614, −3.95430178205688648281340698020, −2.90420436435210811389480878785, −2.30735309602803835186578037678, −1.35781397938448415451142055308, 0, 1.35781397938448415451142055308, 2.30735309602803835186578037678, 2.90420436435210811389480878785, 3.95430178205688648281340698020, 4.33471889531552356021563552614, 5.34443408102195224933515059274, 6.06709869671519499992746204907, 6.62447757568286724337784834863, 7.60402456637185343398224581455

Graph of the ZZ-function along the critical line